Giant Planet Formation: A First Classification of Isothermal Protoplanetary Equilibria

We present a model for the equilibrium of solid planetary cores embedded in a gaseous nebula. From this model we are able to extract an idealized roadmap of all hydrostatic states of the isothermal protoplanets. The complete classification of the isothermal protoplanetary equilibria should improve the understanding of the general problem of giant planet formation, within the framework of the nucleated instability hypothesis. We approximate the protoplanet as a spherically symmetric, isothermal, self-gravitating classical ideal gas envelope in equilibrium, around a rigid body of given mass and density, with the gaseous envelope required to fill the Hill-sphere. Starting only with a core of given mass and an envelope gas density at the core surface, the equilibria are calculated without prescribing the total protoplanetary mass or nebula density. The static critical core masses of the protoplanets for the typical orbits of 1, 5.2, and 30 AU, around a parent star of 1 solar mass are found to be 0.1524, 0.0948, and 0.0335 Earth masses, respectively, for standard nebula conditions (Kusaka et al. 1970). These values are much lower than currently admitted ones primarily because our model is isothermal and the envelope is in thermal equilibrium with the nebula. For a given core, multiple solutions (at least two) are found to fit into the same nebula. We extend the concept of the static critical core mass to the local and global critical core mass. We conclude that the 'global static critical core mass' marks the meeting point of all four qualitatively different envelope regions.Comment: 13 pages, 15 figure

The formation of HD 149026 b

Today, many extrasolar planets have been detected. Some of them exhibit properties quite different from the planets in our solar system and they have eluded attempts to explain their formation. One such case is HD 149026 b. It was discovered by Sato et al. (2005) . A transit-determined orbital inclination results in a total mass of 114 earth masses. The unusually small radius can be explained by a condensible element core with an inferred mass of 67 earth masses for the best fitting theoretical model. In the core accretion model, giant planets are assumed to form around a growing core of condensible materials. With increasing core mass, the amount of gravitationally bound envelope mass increases. This continues up to the so-called critical core mass -- the largest core allowing a hydrostatic envelope. For larger cores, the lack of static solutions forces a dynamic evolution of the protoplanet in the process accreting large amounts of gas or ejecting the envelope. This would prevent the formation of HD 149026 b. By studying all possible hydrostatic equilibria we could show that HD 149026 b can remain hydrostatic up to the inferred heavy core. This is possible if it is formed in-situ in a relatively low-pressure nebula. This formation process is confirmed by fluid-dynamic calculations using the environmental conditions as determined by the hydrostatic models. We present a quantitative in-situ formation scenario for the massive core planet HD 149026 b. Furthermore we predict a wide range of possible core masses for close-in planets like HD 149026 b. This is different from migration where typical critical core masses should be expected.Comment: 6 pages, 2 figures, letter MNRAS accepted 2007 Jan

CoRoT's first seven planets: An overview

The up to 150 day uninterrupted high-precision photometry of about 100000 stars - provided so far by the exoplanet channel of the CoRoT space telescope - gave a new perspective on the planet population of our galactic neighbourhood. The seven planets with very accurate parameters widen the range of known planet properties in almost any respect. Giant planets have been detected at low metallicity, rapidly rotating and active, spotted stars. CoRoT-3 populated the brown dwarf desert and closed the gap of measured physical properties between standard giant planets and very low mass stars. CoRoT extended the known range of planet masses down to 5 Earth masses and up to 21 Jupiter masses, the radii to less than 2 Earth radii and up to the most inflated hot Jupiter found so far, and the periods of planets discovered by transits to 9 days. Two CoRoT planets have host stars with the lowest content of heavy elements known to show a transit hinting towards a different planet-host-star-metallicity relation then the one found by radial-velocity search programs. Finally the properties of the CoRoT-7b prove that terrestrial planets with a density close to Earth exist outside the Solar System. The detection of the secondary transit of CoRoT-1 at the $10^{-5}$-level and the very clear detection of the 1.7 Earth radii of CoRoT-7b at $3.5 10^{-4}$ relative flux are promising evidence of CoRoT being able to detect even smaller, Earth sized planets.Comment: 8 pages, 19 figures and 3 table

The first million years of the Sun: A calculation of formation and early evolution of a solar-mass star

We present the first coherent dynamical study of the cloud fragmentation-phase, collapse and early stellar evolution of a solar mass star. We determine young star properties as the consequence of the parent cloud evolution. Mass, luminosity and effective temperature in the first million years of the proto-Sun result from gravitational fragmentation of a molecular cloud region that produces a cluster of prestellar clumps. We calculate the global dynamical behavior of the cloud using isothermal 3D hydrodynamics and follow the evolution of individual protostars in detail using a 1D radiation-fluid-dynamic system of equations that comprises a correct standard solar model solution, as a limiting case. We calculate the pre-main sequence (PMS) evolutionary tracks of a solar mass star in a dense stellar cluster environment and compare it to one that forms in isolation. Up to an age of 950.000 years differences in the accretion history lead to significantly different temperature and luminosity evolution. As accretion fades and the stars approach their final masses the two dynamic PMS tracks converge. After that the contraction of the quasi-hydrostatic stellar interiors dominate the overall stellar properties and proceed in very similar ways. Hence the position of a star in the Hertzsprung-Russell diagram becomes a function of age and mass only. However, our quantitative description of cloud fragmentation, star formation and early stellar evolution predicts substantial corrections to the classical, i.e. hydrostatic and initially fully convective models: At an age of 1 million years the proto-Sun is twice as bright and 500 Kelvin hotter than according to calculations that neglect the star formation process.Comment: Four pages, accepted for publication in ApJ Letter

Orbital Migration and Mass Accretion of Protoplanets in 3D Global Computations with Nested Grids

We investigate the evolution of protoplanets with different masses embedded in an accretion disk, via global fully three-dimensional hydrodynamical simulations. We consider a range of planetary masses extending from one and a half Earth's masses up to one Jupiter's mass, and we take into account physically realistic gravitational potentials of forming planets. In order to calculate accurately the gravitational torques exerted by disk material and to investigate the accretion process onto the planet, the flow dynamics has to be thoroughly resolved on long as well as short length scales. We achieve this strict resolution requirement by applying a nested-grid refinement technique which allows to greatly enhance the local resolution. Our results from altogether 51 simulations show that for large planetary masses, approximately above a tenth of the Jupiter's mass, migration rates are relatively constant, as expected in type II migration regime and in good agreement with previous two-dimensional calculations. In a range between seven and fifteen Earth's masses, we find a dependency of the migration speed on the planetary mass that yields time scales considerably longer than those predicted by linear analytical theories. This property may be important in determining the overall orbital evolution of protoplanets. The growth time scale is minimum around twenty Earth-masses, but it rapidly increases for both smaller and larger mass values. Significant differences between two- and three-dimensional calculations are found in particular for objects with masses smaller than ten Earth-masses. We also derive an analytical approximation for the numerically computed mass growth rates.Comment: 28 pages, 12 figures. To appear in The Astrophysical Journal vol.586 (March 20, 2003 issue

Planetary migration in evolving planetesimals discs

In the current paper, we further improved the model for the migration of planets introduced in Del Popolo et al. (2001) and extended to time-dependent planetesimal accretion disks in Del Popolo and Eksi (2002). In the current study, the assumption of Del Popolo and Eksi (2002), that the surface density in planetesimals is proportional to that of gas, is released. In order to obtain the evolution of planetesimal density, we use a method developed in Stepinski and Valageas (1997) which is able to simultaneously follow the evolution of gas and solid particles for up to 10^7 yrs. Then, the disk model is coupled to migration model introduced in Del Popolo et al. (2001) in order to obtain the migration rate of the planet in the planetesimal. We find that the properties of solids known to exist in protoplanetary systems, together with reasonable density profiles for the disk, lead to a characteristic radius in the range 0.03-0.2 AU for the final semi-major axis of the giant planet.Comment: IJMP A in prin

An Investigation into the Radial Velocity Variations of CoRoT-7

CoRoT-7b, the first transiting ``superearth'' exoplanet, has a radius of 1.7 R_Earth and a mass of 4.8 M_Earth. Ground-based radial velocity measurements also detected an additional companion with a period of 3.7 days (CoRoT-7c) and a mass of 8.4 M_Earth. The mass of CoRoT-7b is a crucial parameter for planet structure models, but is difficult to determine because CoRoT-7 is a modestly active star and there is at least one additional companion. A Fourier analysis was performed on spectral data for CoRoT-7 taken with the HARPS spectrograph. These data include RV measurements, spectral line bisectors, the full width at half maximum of the cross-correlation function, and Ca II emission. The latter 3 quantities vary due to stellar activity and were used to assess the nature of the observed RV variations. An analysis of a sub-set of the RV measurements where multiple observations were made per night was also used to estimate the RV amplitude from CoRoT-7b that was less sensitive to activity variations. Our analysis indicates that the 0.85-d and 3.7-d RV signals of CoRoT-7b and CoRoT-7c are present in the spectral data with a high degree of statistical significance. We also find evidence for another significant RV signal at 9 days. An analysis of the activity indicator data reveals that this 9-d signal most likely does not arise from activity, but possibly from an additional companion. If due to a planetary companion the mass is m = 19.5 M_Earth, assuming co-planarity with CoRoT-7b. A dynamical study of the three planet system shows that it is stable over several hundred millions of years. Our analysis yields a RV amplitude of 5.04 +/- 1.09 m/s for CoRoT-7b which corresponds to a planet mass of m = 6.9 +/- 1.4 M_Earth. This increased mass would make the planet CoRoT-7b more Earth-like in its internal structure.Comment: 20 pages, 20 figure

Critical Protoplanetary Core Masses in Protoplanetary Disks and the Formation of Short-Period Giant Planets

We study a solid protoplanetary core of 1-10 earth masses migrating through a disk. We suppose the core luminosity is generated as a result of planetesimal accretion and calculate the structure of the gaseous envelope assuming equilibrium. This is a good approximation when the core mass is less than the critical value, M_{crit}, above which rapid gas accretion begins. We model the structure of the protoplanetary nebula as an accretion disk with constant \alpha. We present analytic fits for the steady state relation between disk surface density and mass accretion rate as a function of radius r. We calculate M_{crit} as a function of r, gas accretion rate through the disk, and planetesimal accretion rate onto the core \dot{M}. For a fixed \dot{M}, M_{crit} increases inwards, and it decreases with \dot{M}. We find that \dot{M} onto cores migrating inwards in a time 10^3-10^5 yr at 1 AU is sufficient to prevent the attainment of M_{crit} during the migration process. Only at small radii where planetesimals no longer exist can M_{crit} be attained. At small radii, the runaway gas accretion phase may become longer than the disk lifetime if the core mass is too small. However, massive cores can be built-up through the merger of additional incoming cores on a timescale shorter than for in situ formation. Therefore, feeding zone depletion in the neighborhood of a fixed orbit may be avoided. Accordingly, we suggest that giant planets may begin to form early in the life of the protostellar disk at small radii, on a timescale that may be significantly shorter than for in situ formation. (abridged)Comment: 24 pages (including 9 figures), LaTeX, uses emulateapj.sty, to be published in ApJ, also available at http://www.ucolick.org/~ct/home.htm